SOLUTION: Induction 1^3+3^3+5^3+...+(2n-1)^3 = 2n^4-n^2

Algebra.Com
Question 1092345: Induction
1^3+3^3+5^3+...+(2n-1)^3 = 2n^4-n^2

Answer by ikleyn(52780)   (Show Source): You can put this solution on YOUR website!
.
The key formula is 

 =  +  +  +  . . . +  = 


For the proof see the lesson 
    - Mathematical induction for sequences other than arithmetic or geometric
in this site.


Apply this formula for the sum 

 =  +  +  +  . . . + 
 
=  = .


Then calculate the sum over even natural numbers

 =  +  +  +  . . . +  = 

              = 

              =  = .


Now your sum is equal to   -  =  -  =  =  =  .

It is exactly what you want to prove.   QED.


-------------
To feel yourself freely with these/(with such) problems, you should be familiar with the method of Mathematical induction
and with all associated standard problems around these theme.
There is a group of lessons in this site specially developed for this purpose.
These lessons are

    - Mathematical induction and arithmetic progressions
    - Mathematical induction and geometric progressions
    - Mathematical induction for sequences other than arithmetic or geometric
    - Proving inequalities by the method of Mathematical Induction
    - OVERVIEW of lessons on the Method of Mathematical induction

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic
"Method of Mathematical induction".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.



                    H a p p y   l e a r n i n g  ! !




RELATED QUESTIONS

use the principals of mathematical induction to prove the following statement... (answered by greenestamps)
Using mathematic Induction to prove the next proposition: 1^3 + 3^3 + 5^3 +... +... (answered by ikleyn)
Prove by mathematical induction,1^2+3^2+....(2n+1)^2=((n+1)(2n+1)(2n+3))/3 where 'n' is a (answered by ikleyn)
prove by mathematical induction: 1^5 + 2^5 + 3^5 + ... + n^5 = (1/12)n^2(n+1)^2(2n^2 +... (answered by math_helper)
n+3=2n-1 (answered by rfer,jim_thompson5910)
Mathematical induction How can we prove that : (1 + 1 / 3) (1 + 5 / 4)(1 + 7 /... (answered by ikleyn)
Using mathmatical induction(two step process),, prove the formula... (answered by Jk22)
Please help with this questions, prove by induction that the formulas positive integral... (answered by ikleyn)
5^n+3/25^2n-3=1 (answered by ikleyn,KMST)